Comparing graphs by means of optimal transport has recently gained significant attention, as the distances induced by optimal transport provide both a principled metric between graphs as well as an interpretable description of the associated changes between graphs in terms of a transport plan. As the lack of symmetry introduces challenges in the typically considered formulations, optimal transport distances for graphs have mostly been developed for undirected graphs. Here, we propose two distance measures to compare directed graphs based on variants of optimal transport: (i) an earth movers distance (Wasserstein) and (ii) a Gromov-Wasserstein (GW) distance. We evaluate these two distances and discuss their relative performance for both simulated graph data and real-world directed cell-cell communication graphs, inferred from single-cell RNA-seq data.

翻译：通过最优传输（optimal transport）比较图结构近期受到广泛关注，因为最优传输诱导的距离不仅提供了图间的规范化度量，还能通过传输计划（transport plan）对图间相关变化进行可解释性描述。由于缺乏对称性给传统方法带来挑战，现有图的最优传输距离主要针对无向图发展。本文提出两种基于最优传输变体的有向图比较距离度量：（i）推土机距离（Wasserstein）和（ii）格罗莫夫-瓦瑟斯坦（Gromov-Wasserstein, GW）距离。我们评估了这两种距离的性能，并讨论了其在模拟图数据及从单细胞RNA测序数据推断的真实有向细胞-细胞通信图中的相对表现。